Application of the F-statistic of the Fisher-Snedecor

Article citation info:

PUZDROWSKA, P. Application of the F-statistic of the Fisher-Snedecor distribution to analyze the significance of the effect of changes

in the compression ratio of a diesel engine on the value of the specific enthalpy of the exhaust gas flow. Combustion Engines. 2021,

186(3), 80-88. https://doi.org/10.19206/CE-141346

80 COMBUSTION ENGINES, 2021, 186(3)

Patrycja PUZDROWSKA

Polish Scientific Society of Combustion Engines

Application of the F-statistic of the Fisher-Snedecor distribution to analyze

the significance of the effect of changes in the compression ratio of a diesel engine

on the value of the specific enthalpy of the exhaust gas flow ARTICLE INFO The paper discusses the impact of changes in the compression ratio on the operating parameters of a diesel

engine, e.g. on the temperature of exhaust gases. It presents the construction of the laboratory test stand, on which experimental measurements were realized. It is characterized how the actual changes of the compression

ratio were introduced to the existing engine. The program of experimental investigations taking into account the

available test stand and measurement possibilities was described. A statistical and qualitative analysis of the

obtained measurement results was made. The use of F statistics of the Fisher-Snedecor distribution was

proposed to assess the significance of the effect of compression ratio changes on the specific enthalpy of the

exhaust gas stream. The specific enthalpy of exhaust gases was analysed for one cycle of diesel engine work, determined on the basis of the course of quickly varying temperature of exhaust gases. The results of these

analyses are discussed and the utilitarian purpose of this type of evaluation in parametric diagnostics of piston engines is presented.

Received: 13 July 2021 Revised: 15 August 2021

Accepted: 17 August 2021 Available online: 18 August 2021

Key words: diesel engine, exhaust gas temperature, Fisher-Snedecor decomposition F statistic, diagnosis

This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/)

1. Introduction The compression ratio of the cylinder systems of a pis-

ton engine is a value, which has a significant influence on

the fuel combustion process in these cylinders and on most

of the basic and control parameters of its work, as well as

on the emission of harmful and toxic gaseous components

in the exhaust gases. Decrease or increase of the value

may indicate a state of partial operability or even inopera-

bility of the engine caused, for example, by a loss of tight-

ness or contamination of the cylinder working space. More

and more technologically advanced high-power engines1,

e.g. marine ones, are characterized by limited controllabil-

ity, manifested, for example, by the possibility of cylinder

indicating [14]. Hence the need for parametric diagnosis,

based on quickly changing temperature of exhaust gases,

recorded in the place of standard measurement, which is

required when we talk about marine engines. In this case,

the object of examination are constructional elements limit-

ing working spaces of a marine engine, which, due to re-

quirements of classification societies, has a place of stand-

ard measurement of exhaust gas temperature. Processes

occurring in the inside cylinder spaces are analysed, which

in high-power marine engines are analogous to those in the

analysed single-cylinder engine. The parameter, which is

quickly changing temperature of exhaust gases, can be

a valuable source of diagnostic information, provided that

the diagnosis methodology based on this parameter is

properly prepared. In order to achieve this goal it is im-

portant to choose the right measurement technology, tools

for mathematical processing as well as statistical and con-

tent-related analysis of the obtained results. Conclusions

1 Above 1MW.

based on the F statistic of the Fisher-Snedecor2 distribution

allow us to assess the significance of the influence of one

input factor on one output factor. The input factor in the

discussed case is the structure parameter, i.e. the compres-

sion ratio, and the output factor - the specific enthalpy of

the exhaust gas stream averaged over one engine operation

cycle determined on the basis of quickly varying tempera-

ture of the exhaust gas of a diesel engine.

2. The effect of compression ratio changes

on the operating parameters of a diesel engine One of the most important values characterizing the pis-

ton engine is the compression ratio (determined on the basis

of the engine construction data) or the pressure ratio (de-

termined by measuring the pressure of the working medium

at the beginning and the end of the compression stroke).

This value directly influences the basic engine parameters

(torque, power, specific fuel consumption and efficiency)

and indirectly affects the control parameter, i.e. exhaust

temperature. Thus, the measurement of quickly changing

exhaust gas temperature can be a source of diagnostic in-

formation about decreased or increased compression ratio

resulting from the occurrence of partial serviceability or

unserviceability condition.

2.1. Basic parameters

The design value on which the thermodynamic parame-

ters of the working medium at the end of the compression

stroke depend to a very large degree is the compression

ratio, which tells us how many times the volume of the

working gas decreased in the cylinder during the piston

movement from the bottom dead centre to the top dead

centre [12]:

2 It is also called "F-statistic" for shorter in the rest of the paper.

http://orcid.org/0000-0003-0380-6926

http://www.combustion-engines.eu

Application of the F-statistic of the Fisher-Snedecor distribution…

COMBUSTION ENGINES, 2021, 186(3) 81

ε =Vk+Vs

Vk (1)

where: Vs – volume of cylinder stroke, Vk – volume of

combustion chamber, ε – compression ratio.

On the other hand, the value determined on the basis of

the indicator diagram determined individually for each

cylinder [5] is the pressure ratio, defined by the relation:

ε′ = (p2

p1)

1

𝜅 (2)

where: p1 – pressure at the beginning of the compression

stroke, p2 – pressure at the end of the compression stroke, κ

– isentropy exponent, ε’ – compression (pressure) ratio.

The most important basic and control parameters of en-

gine operation, which depend on the compression ratio,

include those listed below.

The compression ratio should be selected so as to ensure

self-ignition of the fuel injected into the combustion

chamber. It is recommended that the temperature of

compression end be about 200oC higher than the self-

ignition temperature of fuel [11]. This allows for so

called "soft" engine work and easier cold start in various

climatic conditions.

The overall engine efficiency ηo, which is the product of

the theoretical, indicated and mechanical efficiencies,

has a direct effect on the engine operating costs. The

theoretical efficiency of the Seiliger-Sabathe circuit,

which is implemented in a diesel engine, increases with

increasing the value of ε, which can be expressed by the

relation [13]:

ηtS = 1 −1

εκ−1∙

αφκ−1

α−1+ακ(φ−1) (3)

where: ηtS – theoretical efficiency of the Seiliger-

Sabathe cycle, ε – compression ratio, κ – isentropy ex-

ponent, α – isochoric pressure rise ratio, φ – degree of

loading.

In the case of the indicated efficiency ηi, its value

decreases as ε increases. This is because increasing the

value of ε results in an increase in the pressure and tem-

perature of the burning fuel-air mixture. Assuming a

constant coolant temperature value, an increase in com-

bustion temperature results in higher cooling losses. A

similar trend can be observed for the mechanical effi-

ciency ηm (ηm decreases with increasing ε). This is due

to the increase in pressure during combustion, which re-

sults in increased stresses and therefore pressures on the

engine components carrying the forces created by these

pressures. The dynamic loads on the piston-ring-

cylinder system increase the friction losses, which are

the main component of the losses included in ηm.

In the case of diesel engines, the nature of the effi-

ciency curve is similar to that of the spark-ignition en-

gines, but the values of ε are much higher, due to the

differences in the nature of the ignition process: for the

diesel engine from 14 to 23, for the spark-ignition en-

gine from 9 to even 19.

As the compression ratio increases, specific fuel con-

sumption decreases and engine power and torque in-

crease. However, one should also keep in mind the cur-

rent limits of emission of harmful components of ex-

haust gases (CO, HC, NOx, CO2, PM, smoke, etc.), the

amount of which is in direct relation with these parame-

ters [15]. The concentration of toxic components in the

exhaust gas depends primarily on the over-air ratio λ,

and therefore also on the engine load [11].

In order to select an appropriate compression ratio,

many factors should be taken into account, primarily the

type of fuel (its cetane number, or CCAI index – for residu-

al fuels), head and cylinder material, combustion chamber

design, or emission of harmful and toxic components of

exhaust gas [17].

2.2. Temperature of exhaust gases

Assuming that the temperature of the charge at the be-

ginning of the compression stroke is constant, the tempera-

ture of the working medium at the end of this stroke is

determined by the known thermodynamic relation:

T2 = T1 ∙ 𝜀𝜅−1 = T1 ∙ (

V1

V2)𝜅−1

(4)

where: T1 – temperature at the beginning of the compres-

sion stroke, T2 – temperature at the end of the compression

stroke, ε – compression ratio, κ – isentropy exponent, V1 –

volume of working medium at the beginning of the com-

pression stroke, V2 – volume of working medium at the end

of the compression stroke.

However, in determining the end of compression tem-

perature for a known compression ratio (determined from

the pressures) ε', the relation should be used:

T2 = T1 ∙ 𝜀′(𝜅−1) = T1 ∙ (

p2

p1)

𝜅−1

𝜅 (5)

where: p1 – pressure at the beginning of the compression

stroke, p2 – pressure at the end of the compression stroke, ε'

– compression ratio (determined from the pressures – for-

mula 2), κ – isentropy exponent.

So increasing the compression ratio results in an in-

crease in T2. The greater the difference between the mixture

ignition temperature and the temperature of the exhaust gas

leaving the cylinder, the more heat is converted into work.

So the more the medium will be compressed during the

compression stroke (increased ε), the more the medium

must be decompressed to return to the parameters of the

beginning, and thus the gas will do more work and the

exhaust gas temperature will be lower. This is observable

e.g. in the comparison of car engines: petrol (ε = 9–11, texh

= 550–700oC) and Diesel (ε ≈ 20, texh ≈ 400

oC).

Figure 1 shows the dependence of exhaust gas tempera-

ture on the value of compression ratio of a compression

ignition engine [2]. The authors of the publication analyzed

the influence of the injection timing and compression ratio

on the exhaust gas temperature. For all analyzed injection

timing angles, a decrease in exhaust gas temperature with

increasing ε is evident.

In the case of an established diesel engine, a decrease in

compression ratio resulting from increased combustion

chamber volume should be alarming. It may be an indica-

tion of e.g. wear of bearings of the crank system or bearings

of piston rods, which cause shortening of the piston dis-

tance during the compression stroke as a result of the ac-



cumulation of increased releases in the bearings [15]. An-

other reason for the decrease in ε can also be damage to the

connecting rod in the form of bending or buckling of the

connecting rod shaft [6]. The presence of too much soot in

the in-cylinder space, resulting from incomplete combus-

tion of fuel, causes a decrease in the volume of the combus-

tion chamber and therefore an increase in the compression

ratio. Thus, the decrease or increase in exhaust gas tempera-

ture and the deviation of diagnostic measures determined

from it can be a source of information about the occurrence

of a state of partial serviceability or unserviceability.

Fig. 1. Effect of injection timing and compression ratio on exhaust gas temperature; BTDC – before top dead center, POME – palm oil ethyl ester,

BMEP – brake mean effective pressure [2]

3. Research on experimental diesel engine

in the conditions of actually introduced changes

of the compression ratio Identification of the effect of decreased or increased

compression ratio on engine operation is possible through

experimental studies. Simulating the occurrence of the state

of partial serviceability by actually introduced changes of ε

allows to record the selected control parameters in these

conditions. The temperature of the exhaust gas allows us to

determine diagnostic measures and to determine the effect

of the changed compression ratio on the specific enthalpy

of the exhaust gas. However, appropriate mathematical and

statistical processing tools must be kept in mind in order to

maximize the diagnostic informativeness of the exhaust gas

temperature.

3.1. Description of the laboratory test stand

and the measurement devices used

Experiments were carried out on the laboratory test

stand of the single-cylinder, four-stroke Farymann Diesel

engine type D10 (Fig. 2), located in the Laboratory of Ma-

rine Power Plants, Faculty of Mechanical Engineering and

Ship Technology, Gdansk University of Technology. The

basic technical data of the engine are as follows:

nominal power 5.9 kW,

nominal rotational speed 1500 min–1

,

nominal torque 38 Nm,

cylinder diameter 90 mm,

piston stroke 120 mm,

compression ratio 22:1,

volume of cylinder stroke 765 cm3.

During the tests, the following engine control parame-

ters were recorded:

exhaust gas temperature and pressure,

piston top dead centre signal,

load current of the generator (motor)

voltage at the generator armature clamps,

exhaust valve opening signal.

Table 1 shows a summary of the measured control pa-

rameters and the measuring equipment used during the

tests.

A multifunctional measurement and recording module

type DT-9805 from Data Translation was used to record the

quickly changing temperature and pressure of the exhaust

gas as well as the piston top dead center signal, while

Matlab software was used to record the recorded values in

programmer's language. A constant crankshaft speed of

1442–1444 rpm was maintained throughout the test. The

sampling frequency was approximately 7000 Hz. The pre-

sented test results are the average of 90 consecutive meas-

urements recorded under the same engine operating condi-

tions determined by the engine load, crankshaft speed and

ambient parameters. During the tests the engine was burn-

ing MGO type marine fuel. In order to obtain a lower com-

pression ratio value, an additional structural element was

designed and manufactured, increasing the combustion

chamber volume (Fig. 2).

Tab. 1. The parameters of the Farymann D10 single cylinder diesel engine recorded on the laboratory test stand

Item Parameter Measuring device Unit Measurement range

1. Exhaust gas temperature – Tsp Grounded type K thermocouple with the

junction of external diameter of 0.5 mm, made from inconell

oC 0–1000

2. Exhaust gas pressure in the exhaust channel – psp Optical pressure sensor – Optrand C12296 V

0–689475.73 Pa (0–100 psi),

sensitivity 6.01·10–8 V/Pa

(41.43 mV/psi)

3. Engine speed (angular position oCA) – n

Top dead center – TDC

Induction engine speed sensor and TDC

sensor min–1 0–3000

4. Load Current of the generator (armature) – Itw Electric current meter A 0–15

5. Voltage at the armature terminals– Utw Voltmeter V 0–250

6. Exhaust valve opening signal Gap type opto-isolator with a comparator

LM393 V

mm 0–5

10 (gap)



a) b)

Fig. 2. a) diagram of the laboratory test stand with the fitting spots of the sensors marked: 1 – Farymann D10 engine, 2 – engine speed and TDC sensor,

3 – exhaust valve opening sensor, 4 – A/C converter, 5 – recorder, 6 – analysis software, 7 – component to increase the volume of the combustion cham-ber, 8 – pressure sensor, 9 – water cooled thermocouple, 10 – exhaust gas channel, A – intake air, B – exhaust gas, C – fuel line; b) view of the laboratory

test stand with the fitting spots of the sensors of the recorded parameters marked: 2 – engine speed and TDC sensor, 3 – exhaust valve opening sensor, 7 –

component to increase the volume of the combustion chamber with a pressure sensor for the medium inside the combustion chamber, 8 – pressure sensor, 9 – water cooled thermocouple, A – intake air, C – fuel line

3.2. Experimental research program

In the research realised by the author of this paper, the

main aim was to determine the diagnostic informativeness

of the parameter, i.e. quickly changing temperature of the

exhaust gas of a diesel engine, depending on the actually

introduced changes in selected parameters of its construc-

tional structure. Therefore, it was decided to use a static,

randomized and complete experiment plan [4]. This paper

presents part of the results of a larger program of research

and statistical analysis undertaken, namely the evaluation of

the significance of the effect of the compression ratio on the

specific enthalpy of the exhaust gas of a single-cylinder

diesel research engine. On this basis it is possible to transfer

the results of laboratory tests to full-size marine engines,

for their diagnosis (diagnostic conclusions).

The experimental research was carried out on the Fary-

mann Diesel engine type D10. The research object of ther-

mo-flow processes was defined by constructional elements

limiting the cylinder working space, as well as by the chan-

nels of inlet air and outlet exhaust gases. In the research and

analyses carried out, the processes occurring in the intra-

cylinder space were not subjected to, and only the signal of

quick-changing temperature of exhaust gases recorded in

the exhaust gas channel was examined. The input parame-

ters, output parameters, disturbing factors and constants are

summarized in Fig. 3. In case of the applied model, the

tested engine was treated as an object of diagnosis on the

assumption that the set of U forcing is the parameters of

engine operation (load, torque, rotational speed), which do

not change. However, the state parameters X (of the con-

structional structure) are variable – in the discussed case, it

is the degree of compression of the engine. The output

parameter Y1, which is the exhaust gas pressure, is im-

portant for determining the exhaust gas enthalpy flow of the

engine Hspal. Diagnostic measures Y2 were determined on

the basis of the recorded signal of quickly changing tem-

perature of exhaust gases, however, in the present paper we

discuss only the specific enthalpy of the exhaust gas stream

averaged over one cycle of diesel engine operation – hśr.

The evaluation of the effect of variable engine load as

an input parameter on selected diagnostic measures was

described by the author in the article [7]. The variable value

of the inlet air flow area, as a simulation of the loss of puri-

ty of the inlet air duct, was analyzed in the paper [8].

Meanwhile, the evaluation of the influence of the structure

parameter, which is the reduced injector opening pressure

(spring relaxation), as well as the analysis of the influence

of the mentioned input parameters (load, inlet air flow area,

injector opening pressure, and compression ratio) on all

analyzed diagnostic measures (described in [7]) are in prep-

aration for publication.

In steady-state operation of a diesel engine, three char-

acteristics of its operation are possible [1]. In the carried out

research, the speed, regulator characteristic was used, with

the rotational speed changing in the range of astatic opera-

tion of the regulator. In this case, the engine is loaded by

increasing the generated effective torque (increasing the

fuel dose per engine cycle), at a constant value of crank-

shaft rotational speed.



Fig. 3. Simplified physical model of a diagnostic subject for an elimination experimental test realized according to a randomized complete plan

Fig. 4. The characteristic – regulator load variation of Farymann Diesel engine type D10 as a function of crankshaft rotational speed used during testing

(variable fuel dose per engine cycle at crankshaft rotational speed stabilized by the regulator), nNR – rotational speed regulator setting, nRZ – real rotational speed

During the experimental research carried out on Fary-

mann Diesel engine type D10 the measurements were real-

ized for 3 points of work according to the regulator charac-

teristics (Fig. 4). In the discussed stage of research and

mathematical processing and statistical analysis, changes

were made in the structure parameter, i.e. combustion

chamber volume, which was represented in the values of

engine compression ratio. The reference value for the en-

gine being tested was εREF = ε1 = 22:1, while the reduced

value was ε2 = 21:1, due to design limitations. Reduction of

compression ratio was realized by application of additional

structural element, increasing the volume of combustion

chamber by about 0.125–10–5

m3 (ΔVk), with initial (refer-

ence) volume of combustion chamber Vk1 = 3.787·10–5

m3

and volume of cylinder stroke Vs = 79.5·10–5

m3.

During the diagnostic testing of an engine, at the time of

established operation, diagnostic parameters are determined

from among its output parameters, which react more strong-

ly to changes in the values of structural parameters than to

changes in the values of input parameters, forcing the real-

ized working process. The basic condition for selecting

adequate diagnostic parameters is that the sensitivity of the

output parameter to the structure parameter is much higher

than its sensitivity to the input parameter [3]. Comparing

the sensitivities of multiple control parameters, given in

different units of measurement, requires taking relative

values of input, output and structure parameters for this

purpose. There are many methods to assess the significance

of the influence of the forcing parameters of the analyzed

physical process on its course, but in this study a static



randomized complete program was used [4], and the as-

sumed null hypothesis is that there is no influence of the

input factor on the output factor. The influence of an input

factor is considered significant when the calculated value of

the used statistic is equal to or greater than the critical val-

ue, given in the tables for the assumed value of the signifi-

cance level α and the number of degrees of freedom f = n – 1.

It was considered best to use the F statistic of the Fisher-

Snedecor distribution [4, 9, 16], as the conditions for using

one-sided parametric tests were met. In the conducted re-

search, it was assumed that the measurement results of all

control parameters can be modeled as random variables

with a normal distribution, a defined expected value and

variance, which is a measure of the spread of measurement

results around the expected value. The variances of the

random variables were also assumed to be equal or close in

value, and the parametric tests for variance used are with

a one-sided critical area. The possibility of making an error

of the first type associated with the assumed significance

level α, i.e., the probability of rejecting the null hypothesis

when it is true, and the possibility of making an error of the

second type, i.e., accepting the null hypothesis when it is

false, of β = 1 – α were considered.

Table 2 presents the matrix of the experimental research

program, in this case the randomized static plan, allowing

the evaluation of the significance of the influence of the

input factor considered in the diagnostic tests of the engine

– the compression ratio of the engine realized within the

specified range of variability, on the diagnostic parameter

determined in these tests (the output factor), which is the

specific enthalpy of the exhaust gas stream within the range

of one working cycle – hspal.

Table 2. Experimental research program matrix – static randomized com-

plete plan

Level of

input factor

Number of experience

1 ... 4

ε1 hspal11 ... hspal41

ε2 hspal12 ... hspal42

The test (calculation) value of the F statistic of the Fish-

er-Snedecor distribution is determined from the following

relationship [4]:

F =∑ ni(hi−h)

2(n−p)

p

i=1

[∑ ∑ (hi−h)2−∑ ni(hi−h)

2pi=1

qj=1

pi=1

](p−1) (4)

where: ni – number of measurements of specific enthalpy at

a given level, n – total number of measurements, hi – aver-

age specific enthalpy from the measurements in the i line, h

– average specific enthalpy from all measurements, hij – the

value of the j specific enthalpy at the i level, p – number of

levels of variation of the input medium (compression ratio).

The calculated value of Fobl statistic is compared with its

critical value Fkr determined from the statistical table, for

the assumed significance level α and for the determined

numbers of degrees of freedom defined for the numerator

and denominator: f1 and f2. If the determined value of Fobl is

greater than or equal to the critical value of Fkr, then the

influence of the factor under study is to be considered sig-

nificant. Otherwise, the input factor under study is consid-

ered to have no significant effect on the output factor within

the range of variation under study and at the assumed sig-

nificance level. Additionally, once a factor is considered

significant (Fobl > Fkr), it is possible to compare the differ-

ence ΔF = Fobl – Fkr for individual diagnostic measures.

This allows us to assess which diagnostic measure is more

strongly influenced by the input parameter (e.g. structure) –

the higher the ΔF, the greater the influence [10].

In order to get the value of the Fobl statistic and the dif-

ference ΔF, and thus to answer the question regarding the

significance of the influence of selected input factors (struc-

ture) of the engine on the diagnostic measure, i.e. average

specific enthalpy of exhaust gases within one cycle, we had

to proceed in accordance with the scheme of research reali-

zation – Fig. 5.

Fig. 5. Steps in determining the F-statistic of the Fisher-Snedecor distribu-

tion for diagnostic purposes



3.3. Statistical and qualitative analysis of the results

achieved

The values of the output factor, i.e. the specific enthalpy

of the exhaust gas stream during one cycle of engine opera-

tion, for three states of the engine steady-state load, are

presented in Tables 3a–c. The points P1, P2 and P3 deter-

mined by these parameters result from the applied control

characteristics of the engine operation (Fig. 4).

The following zero hypothesis was used to determine

the value of the Fobl statistic:

H0: the value of compression ratio has no effect on the

value of specific enthalpy of the exhaust gas stream

averaged over one engine operating cycle (SII2 = SI

2)

On the basis of the numerical data summarized in Ta-

bles 3a–c and the applied level of significance α = 0.05 and

the assumption of the right-side critical area, the specific

enthalpy of the exhaust gas stream within one engine cycle

was determined for each measurement point (Pi), the num-

ber of degrees of freedom for the numerator and denomina-

tor (f1 = 1 and f2 = 6). Then, the critical value of the statistic

Fkr = F(0.05;1;6) = 5.9874 was read from statistical tables

[4] and the values of Fobl were calculated, whose values are

presented in Table 4. In the case of all analysed characteris-

tic points the condition: Fobl > Fkr (positive ΔF) is fulfilled,

so the zero hypothesis should be rejected and it should be

assumed in further diagnostic tests that in the considered

range of variation of the engine load and the value of the

compression ratio it (ε) has a significant influence on the

specific enthalpy of the exhaust gases determined within

one cycle of the diesel engine work.

Table 3a. Value (average within one working cycle) of exhaust gas enthalpy hśr for variable values of compression ratio in point P1

P1 (432 W; 5.1 A; 72 V)

Point [–] Number of experience

1 2 3 4 yi

ε1 22 12.1299 12.1986 12.2763 12.2176 12.2056

ε2 21 13.3422 13.0606 12.9157 12.8373 13.0389

Table 3b. Value (average within one working cycle) of exhaust gas enthalpy hśr for variable values of compression ratio in point P2

P2 (768 W; 6.8 A; 96 V)


1 2 3 4 yi

ε1 22 13.8992 13.6571 13.4404 13.3947 13.5979

ε2 21 14.4089 14.5475 14.0739 14.2577 14.3220

Table 3c. Value (average within one working cycle) of exhaust gas enthalpy hśr for variable values of compression ratio in point P3

P3 (1200 W; 8.5 A; 120 V)


1 2 3 4 yi

ε1 22 15.4712 15.4883 15.3075 15.3299 15.3992

ε2 21 16.4605 15.8383 15.7786 15.8411 15.9796

Table 4. Value of statistic Fobl and (ΔF = Fobl – Fkr) for a reduced compression ratio and its effect on hśr

Points according to the regulator characteristics Fobl i (ΔF = Fobl – Fkr) for hśr [kJ/kg]

P1 (432 W; 5.1 A; 72 V) 52.34

(46.35)

P2 (768 W; 6.8 A; 96 V) 22.12

(16.13)

P3 (1200 W; 8.5 A; 120 V) 11.99

(6.00)

Fig. 6. Effect of the compression ratio (ε) on the value of the average within one cycle of the exhaust gas enthalpy hśr for diesel engine loads according to

the regulator characteristics



It can also be seen (Table 4) that at point P2 the value

of ΔF is almost 3 times higher and at point P1 almost 8

times higher than at P3. It is therefore possible to conclude

that for lower loads of the diesel engine, the effect of com-

pression ratio on hśr is greater than for higher loads.

In developing a methodology for diagnosing a diesel

engine based on quickly changing exhaust gas temperature,

it is important to consider a number of factors. First, from

the output parameters obtained during engine testing, it is

necessary to select those that contribute the greatest amount

of diagnostic information about the engine condition. Next,

an appropriate tool should be found to assess the signifi-

cance of the effect for parameters presented in different

units of measurement. After the selection of an appropriate

test program and a method of statistical analysis, a scheme

for carrying out the tests should be developed. It should

take into account, among others, mathematical processing

(correction) of the obtained measurement signal by means

of individually selected filters. At the final stage – during

the assessment of the significance of the considered input

factors on the output, both single and multivariate analysis

is possible. In the case of the discussed analysis of variance,

the determination of the value ΔF = Fobl – Fkr also allows us

to assess how large is the influence of the input factor on

the output. On the basis of carried out experimental re-

search and statistical analysis of the obtained results, it is

possible to assess the influence of the compression ratio on

the value of one cycle average enthalpy of exhaust gas hśr

determined on the basis of quickly changing temperature of

exhaust gas (Fig. 6). Thanks to obtaining the results of

similar analysis also for other diagnostic measures calculat-

ed on the basis of this parameter and for other structure

parameters, it is possible to develop the diagnosis method-

ology based on this temperature. This is particularly im-

portant in the diagnostics of engines with limited controlla-

bility, in which the measurement of the exhaust gas temper-

ature is the main source of information on the technical

condition of the working spaces of the diesel engine. The

conducted research shows that a decrease in the compres-

sion ratio (resulting from the occurrence of partial servicea-

bility or unserviceability) influences significantly the in-

crease in the average value within one cycle of the exhaust

gas enthalpy. Additionally, performing a statistical analysis

based on the value of ΔF allows to determine the signifi-

cance of the effect of ε on hśr.

4. Comments and final conclusions The compression ratio ε is an important quantity charac-

terizing the operation of a diesel engine. It can be deter-

mined either as a design parameter or by analysis of the

indicator diagram. In both cases the engine user should be

alerted by its increase or decrease, showing the occurrence

of a state of partial serviceability or unserviceability. The

measurement of quick changing temperature of exhaust

gases allows to determine the influence of the change of

input value ε on diagnostic measure, that is hśr. It is possible

due to application of statistical tool that is F statistic of

Fisher-Snedecor distribution. In the investigated range of

engine load variation the significant influence of ε on hśr

was determined. The applied research and analytical meth-

od was found to be effective for evaluating the influence of

significance of changes of the input factor – structure (ε) on

the output parameter (hśr). It is also possible to determine

other diagnostic measures from the quickly changing ex-

haust gas temperature and determine the effect of ε on them

by using F statistics.

Nomenclature

BMEP brake mean effective pressure

BTDC before top dead center

F test value of the statistic of the Fisher-Snedecor

POME palm oil ethyl ester

ε compression ratio

κ isentropy exponent

ηm mechanical efficiency

ηtS theoretical efficiency of the Seiliger-Sabathe cycle

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Documents

Application of the F-statistic of the Fisher-Snedecor